The midfielder has one-year left on his current contract and had rejected the Goodison Park outfit's initial offer of a new deal several weeks ago.
Barmby's summer took a positive twist when he was brought into Kevin Keegan's England squad in place of Jamie Redknapp, who was ruled out of travelling to Belgium and Holland due to a knee injury.
He impressed as a substitute against Ukraine and could become a key asset as England aim to progresses in the tournament.
Everton chief executive Michael Dunford said: "When he returns from Euro 2000 we'll sit down and ascertain whether he wants to stay and what financial package he's seeking."
